A note on the existence of fractional f-factors in random graphs

Abstract

Let G = G _n,p be a binomial random graph with n vertices and edge probability p = p(n), and f be a nonnegative integer-valued function defined on V(G) such that $d_G^h (x) = \sum\limits_{x \in e} {h(e)}$ for every x ∈ V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0, 1] so that for each vertex x, we have d ^h _G (x) = f(x), where 0 < a ≤ f(x) ≤ b < np − 2√nplogn is the fractional degree of x in G. Set E _h = {e: e ∈ E(G) and h(e) ≠ 0}. If G _h is a spanning subgraph of G such that E(G _h ) = E _h , then G _h is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph G _n,p with $p \geqslant n^{ - \tfrac{2} {3}}$, almost surely G _n,p contains an fractional f-factor.